Generalisation of the Hammersley-Clifford Theorem on Bipartite Graphs

The Hammersley-Clifford theorem states that if the support of a Markov random field has a safe symbol then it is a Gibbs state with some nearest neighbour interaction. In this paper we generalise the theorem with an added condition that the underlying graph is bipartite. Taking inspiration from "Gibbs Measures and Dismantlable Graphs" by Brightwell and Winkler we introduce a notion of folding for configuration spaces called strong config-folding proving that if all Markov random fields supported on $X$ are Gibbs with some nearest neighbour interaction so are Markov random fields supported on the 'strong config-folds' and 'strong config-unfolds' of $X$.Comment: 27 pages, 7 figures; Typos corrected and some notation has been change

Hom complexes and homotopy theory in the category of graphs

We investigate a notion of $\times$-homotopy of graph maps that is based on the internal hom associated to the categorical product in the category of graphs. It is shown that graph $\times$-homotopy is characterized by the topological properties of the \Hom complex, a functorial way to assign a poset (and hence topological space) to a pair of graphs; \Hom complexes were introduced by Lov\'{a}sz and further studied by Babson and Kozlov to give topological bounds on chromatic number. Along the way, we also establish some structural properties of \Hom complexes involving products and exponentials of graphs, as well as a symmetry result which can be used to reprove a theorem of Kozlov involving foldings of graphs. Graph $\times$-homotopy naturally leads to a notion of homotopy equivalence which we show has several equivalent characterizations. We apply the notions of $\times$-homotopy equivalence to the class of dismantlable graphs to get a list of conditions that again characterize these. We end with a discussion of graph homotopies arising from other internal homs, including the construction of `$A$-theory' associated to the cartesian product in the category of reflexive graphs.Comment: 28 pages, 13 figures, final version, to be published in European J. Com

A Brightwell-Winkler type characterisation of NU graphs

In 2000, Brightwell and Winkler characterised dismantlable graphs as the graphs $H$ for which the Hom-graph ${\rm Hom}(G,H)$, defined on the set of homomorphisms from $G$ to $H$, is connected for all graphs $G$. This shows that the reconfiguration version ${\rm Recon_{Hom}}(H)$ of the $H$-colouring problem, in which one must decide for a given $G$ whether ${\rm Hom}(G,H)$ is connected, is trivial if and only if $H$ is dismantlable. We prove a similar starting point for the reconfiguration version of the $H$-extension problem. Where ${\rm Hom}(G,H;p)$ is the subgraph of the Hom-graph ${\rm Hom}(G,H)$ induced by the $H$-colourings extending the $H$-precolouring $p$ of $G$, the reconfiguration version ${\rm Recon_{Ext}(H)}$ of the $H$-extension problem asks, for a given $H$-precolouring $p$ of a graph $G$, if ${\rm Hom}(G,H;p)$ is connected. We show that the graphs $H$ for which ${\rm Hom}(G,H;p)$ is connected for every choice of $(G,p)$ are exactly the ${\rm NU}$ graphs. This gives a new characterisation of ${\rm NU}$ graphs, a nice class of graphs that is important in the algebraic approach to the ${\rm CSP}$-dichotomy. We further give bounds on the diameter of ${\rm Hom}(G,H;p)$ for ${\rm NU}$ graphs $H$, and show that shortest path between two vertices of ${\rm Hom}(G,H;p)$ can be found in parameterised polynomial time. We apply our results to the problem of shortest path reconfiguration, significantly extending recent results.Comment: 17 pages, 1 figur